Can abstractive summarization be achieved using neural networks?
The ability to re-frame summarization as a problem for ANN is rather dependent on what kind of output you're looking for: you mentioned 'salient parts of the text'.

One possibly is to use a deep learning approach that first chunks together words that belong in the same phrase as a single 'feature'.

Another possibility is to identify both key words and relations between them. Here is some previous work on using neural nets for relational learning.